upper bound

Let $$X$$ be an ordered set, and let $$A$$ be a subset of $$X$$. Then an element $$u$$ of $$X$$ is said to be an upper bound of $$A$$ if $$u \geq a$$ for every $$a\in A$$. If in addition $$u \leq v$$ for every upper bound $$v$$ of $$A$$, then $$u$$ is said to be a least upper bound or supremum of $$A$$.

The terms lower bound, greatest lower bound, and infimum are defined analogously.